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Using the family symmetry, in the neutrino mass matrix we remove the Yukawacoupling (arising in the Dirac type mass between the heavy neutrinos and lightlepton doublets) dependence in the double seesaw mechanism so that it isdirectly proportional to the mass matrix $m^{(nn)}$ of heavy Majorananeutrinos. The family symmetry is supposed to be broken spontaneously at highenergy scale so that the neutrino mass matrix is given by the family symmetryat high energy scale. With the permutation symmetry $S_3$, we note a variety ofpossible mass hierarchies arising distinctly in neutrinos, charged leptons,$Q_{\rm em}=-\frac13$ quarks, and $Q_{\rm em}=\frac23$ quarks. Distinguishingthese hierarchies, we obtain a relation between the CKM angles and the MNSangles. Finally, we comment on the approximate relation $\theta_{\rmsol}+\theta_c\simeq \frac{\pi}{4}$.Comment: LaTeX file of 26 pages including 2 figure

the journal of high energy physics/the journal of high energy physics

Quantum numbers of heavy neutrinos, tri-bi-maximal mixing through double seesaw with permutation symmetry, and comment on θsol+ θcsimeq π/4